The OEIS is supported by the many generous donors to the OEIS Foundation.
%I #58 Jan 11 2023 06:41:08
%S 0,1,1,0,1,0,1,1,1,1,0,1,0,1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,1,1,1,1,1,1,
%T 1,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,
%U 1,1,1,1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N Triangle read by rows where all entries in every even row are 1's and the entries in every odd row alternate between 0 (start/end) and 1.
%C Row sums are equal to n for even n and (n-1)/2 for odd n; or A065423(n+1).
%F T(n, k) = 1/2 + (1/2)*(-1)^(n*(k+1)), for n >= 1 and 0 <= k <= n-1.
%F T(n, k) = (2^n - 2^(n-k-1) - 2^k) mod 3, for n >= 1 and 0 <= k <= n-1.
%F T(n, k) = A358125(n, k) mod 3, for n >= 1 and 0 <= k <= n-1.
%e Triangle begins:
%e n\k 0 1 2 3 4 5 6 7 8 9 ...
%e 1 0;
%e 2 1, 1;
%e 3 0, 1, 0;
%e 4 1, 1, 1, 1;
%e 5 0, 1, 0, 1, 0;
%e 6 1, 1, 1, 1, 1, 1;
%e 7 0, 1, 0, 1, 0, 1, 0;
%e 8 1, 1, 1, 1, 1, 1, 1, 1;
%e 9 0, 1, 0, 1, 0, 1, 0, 1, 0;
%e 10 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
%e ...
%e Formatted as a symmetric triangle -- regular hexagram pattern with 0's at the centers formed by connecting all 1's:
%e .----------------------------------------------.
%e | k=0 1 2 3 4 5 |
%e |-----------------------/---/---/---/---/--./ |
%e ------- / / / / / |
%e | n=1 | 0 / / / / /| |
%e ------- / / / / | 6 |
%e | 2 | 1---1 / / / / |/ |
%e ------- \ / / / / / |
%e | 3 | 0 1 0 / / / /| |
%e ------- / \ / / / | 7 |
%e | 4 | 1---1---1---1 / / / |/ |
%e ------- \ / \ / / / / |
%e | 5 | 0 1 0 1 0 / / /| |
%e ------- / \ / \ / / | 8 |
%e | 6 | 1---1---1---1---1---1 / / |/ |
%e ------- \ / \ / \ / / / |
%e | 7 | 0 1 0 1 0 1 0 / /| |
%e ------- / \ / \ / \ / | 9 |
%e | 8 | 1---1---1---1---1---1---1---1 / / |
%e ------- \ / \ / \ / \ / / |
%e | 9 | 0 1 0 1 0 1 0 1 0 /| |
%e ------- / \ / \ / \ / \ | . |
%e | 10 | 1---1---1---1---1---1---1---1---1---1 | . |
%e ------- | . |
%p T := n -> local k; seq(1/2 + 1/2*(-1)^(n*(k + 1)), k = 0 .. n - 1); # formula 1
%p seq(T(n), n=1..16); # print first 16 rows of formula 1.
%o (PARI) T(n,k) = bitnegimply(1,n) || bitand(1,k); \\ _Kevin Ryde_, Dec 21 2022
%Y Cf. A358125, A065423 (row sums).
%K nonn,easy,tabl
%O 1,1
%A _Ambrosio Valencia-Romero_, Dec 20 2022